Joan wants to start an IRA that will have $250,000 in it when she retires in 21 years. How much should she invest annually in her IRA to do this if the interest is 16% compounded quarterly?
To calculate the annual investment Joan should make in her IRA, we can use the future value of an annuity formula. The formula is:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
Where:
FV = Future Value of the IRA ($250,000)
P = Annual investment amount
r = Annual interest rate (16% or 0.16)
n = Number of times interest is compounded per year (quarterly, so n = 4)
t = Number of years (21)
Using these values, we can plug them into the formula and solve for P:
250,000 = P * [(1 + 0.16/4)^(4*21) - 1] / (0.16/4)
Now let's calculate it step by step:
Step 1: Calculate the quarterly interest rate:
Quarterly Interest Rate (r/n) = 0.16 / 4 = 0.04
Step 2: Calculate the number of compounding periods:
Number of Compounding Periods (nt) = 4 * 21 = 84
Step 3: Calculate the numerator of the formula:
Numerator = (1 + 0.04)^84 - 1
Step 4: Calculate the denominator of the formula:
Denominator = 0.16 / 4
Step 5: Calculate the annual investment amount (P):
P = 250,000 * Denominator / Numerator
Now let's substitute the values and calculate P.
Numerator = (1 + 0.04)^84 - 1 = 9.64634287
Denominator = 0.16 / 4 = 0.04
P = 250,000 * 0.04 / 9.64634287
P ≈ $1,040.31
Therefore, Joan should invest approximately $1,040.31 annually in her IRA to achieve $250,000 when she retires in 21 years with an interest rate of 16% compounded quarterly
To find out how much Joan should invest annually in her IRA, we can use the formula for calculating the future value of an ordinary annuity:
FV = P ((1 + r/n)^(nt) - 1) / (r/n)
Where:
- FV is the future value of the annuity ($250,000)
- P is the annual payment (which we need to find)
- r is the interest rate per compounding period (16% or 0.16)
- n is the number of compounding periods per year (quarterly, so n = 4)
- t is the total number of years (21)
Plugging in the values, the formula becomes:
250,000 = P ((1 + 0.16/4)^(4*21) - 1) / (0.16/4)
Simplifying this equation, we have:
250,000 = P (1.04^84 - 1) / 0.04
To solve for P, we can isolate it by multiplying both sides of the equation by 0.04 and then dividing by (1.04^84 - 1):
P = 250,000 * 0.04 / (1.04^84 - 1)
Using a calculator, we can compute this value to determine how much Joan should invest annually in her IRA.
A = P(1+r/n)^nt
250000 = P(1.04)^4*21
P = 250000/1.04^84
P = 9271.28
Solve the problem. Round to the nearest cent.
Joan wants to have $250,000 when she retires in 27 years. How much should she invest annually in her annuity to do this if the interest is 7% compounded annually?
A) $1861.10
B) $3356.43
C) $2672.15
D) $937.86